Elliptical Distributions Vadym Omelchenko Examples of the Elliptical

Elliptical Distributions Vadym Omelchenko

Examples of the Elliptical Distributions � Normal Distribution � Laplace Distribution � t-Student Distribution � Cauchy Distribution � Logistic Distribution � Symmetric Stable Laws

Examples of the Elliptical Distributions

Bivariate Normal Distribution

Multivariate Normal Distribution with correlation equal to 0. 7

� The further ρ from zero the more evident ellipticity of the map, when observing it from above. When ρ=0 then the map has the spherical form.

Laplace bivariate distribution

Definition of elliptical distributions � The random vector is said to have an elliptical distribution with parameters vector and the matrix if its characteristic function can be expressed as � for some scalar function and where and Σ are given by

Characteristic Function of the Symmetric Stable Distributions

� If X has an elliptical distribution, we write X where is called characteristic generator of X and hence, the characteristic generator of the multivariate normal is given by � The random vector X does not, in general, possess a density but if it does, it will have the form For some non-negative function called density generator and for some constant called normalizing constant.

Alternative Denoting of the Elliptical Distributions �X where generator assuming that is the density exists.

Mean and Covariance Properties � If X it will be � If then if the mean exists then the variance matrix exists, it will be � That is, the matrix Σ coincides with the covariance matrix up to the constant.

Mean and Covariance Properties � Examples of the distributions that don’t have mean nor variance: � All stable distributions whose index of stability is lower than 1, e. g. Cauchy or Levy.

Mean and Covariance Properties � Let � X . Then Corollary. Let X , let B be a matrix and . Then Hence marginal distributions of elliptical distributions are elliptical distributions.

Convolutional Properties �

Convolutional Properties � Hence follows that the sum of elliptical distribution is an elliptical distribution. This property is very important when we deal with portfolio of assets, represented by sum.

Basic Properties of the Elliptical Distributions � 1. Elliptical distributions can be seen as an extension of the Normal distribution � 2. Any linear combination of elliptical distributions is an elliptical distribution � 3. Zero correlation of two normal variables implies independence only for Normal distribution. This implication does not hold for any other elliptical distribution.

Basic Properties of the Elliptical Distributions � 4. X with rank(Σ)=k if X has the same distribution as � Where (radius ) and is uniformly distributed on unit sphere surface in and A is a (k×p) matrix such that

Basic Properties of the Elliptical Distributions � As it was mentioned above, if the elliptically distributed function has a density then it is of the form: � The condition guarantees that is a density generator.

Examples of density generators

Construction of a density function with infinite variance

Financial Application Expected Shortfall � The expected shortfall (or tail conditional expectation) is defined as follows: � and can be interpreted as the expected worse losses.

Expected Shortfall the familiar normal distribution N(μ, ), � with mean μ and variance , it was noticed by Panjer (2002) that: � For �

Generalization of the Previous Formula � Suppose that g(x) is a non-negative function for any positive number, satisfying the condition that: � Then g(x) can be a density generator of a univariate elliptical distribution of a random variable X

Generalization of the Formula for the Normal Law � The density of this function has the form: � where c is a normalizing constant. � If X has an elliptical distribution then � � Has a standard elliptical distribution (spherical)

Generalization of the Formula for the Normal Law � The distribution function of Z has the form: � With mean 0 and variance equal to

Generalization of the Formula for the Normal Law � Define the function G(x) which we will call cumulative generator.

Theorem 1 � Let X and G be the cumulative generator. Under condition (*), the tail conditional expectation of X is given by � � Where λ is expressed as

Examples � 1. For Cauchy distribution the TCE doesn’t exist. Because it doesn’t satisfy conditions of theorem

Logistic Distribution �

Sums of Elliptical Risks � Suppose X is the vector of ones with dimension n. Define

Theorem 2 � The � This TCE can be expressed as theorem holds as a result of convolution properties of the elliptical distributions and the previous theorem.

Sums of Elliptical Risks Suppose X is the vector of ones with dimension n, and � Then the contribution of overall risk can be expressed as: � to the

Skewed Elliptical Distributions � All elliptical distributions belong to this family. � All stable distributions belong to this family. � The density of the skewd Normal Distribution has a form:

Skewed Elliptic Distributions Stable Distributions Skewed stable

Literatura � 1. TAIL CONDITIONAL EXPECTATIONS FOR ELLIPTICAL � DISTRIBUTIONS � Zinoviy M. Landsman* and Emiliano A. Valdez† � 2. CAPM and Option Pricing with Elliptical Distributions, Hamada M, Valdez. � 3. Handbook of Heavy Tailed Distributions in Finance, Eds S. T. Rachev